Houston posted a 4-3 aggregate goal victory in the two-match series with Cummings becoming the hero for the second straight game.
Cummings produced a goal in stoppage time to bring about a 2-2 draw in the first leg of the conference semifinals.
Regulation time in Wednesday's game ended with the teams tied, which brought about an overtime in which the full 30 minutes were played.
The goal by Cummings came just prior to the midway point of that overtime when he received a header from Cam Weaver and drove a shot into the right corner of the net.
New York was stunningly eliminated from the playoffs after winning the Eastern Conference regular-season title and finishing with the best record in MLS.
Houston fell behind 2-0 in the first leg before coming back to tie it. Bradley Wright-Phillips gave New York the lead in Wednesday's game by scoring in the 23rd minute, but Houston deadlocked the contest 13 minutes later on a score from Brad Davis.
LGBT community has 'bullied the American people': Bachmann
Beautician charged with giving client fatal silicone butt injection